Multi-Arm Cooperating Robots: Dynamics and Control

297 Pages·2006·3.97 MB·English

by  Zivanovic M. D.

Multi-Arm Cooperating Robots
International Series on
MICROPROCESSOR-BASED AND
INTELLIGENT SYSTEMS ENGINEERING
VOLUME 30
Editor
Professor S. G. Tzafestas, National Technical University of Athens, Greece
Editorial Advisory Board
Professor C. S. Chen, University of Akron, Ohio, U.S.A.
Professor T. Fokuda, Nagoya University, Japan
Professor F. Harashima, University of Tokyo, Tokyo, Japan
Professor G. Schmidt, Technical University of Munich, Germany
Professor N. K. Sinha, McMaster University, Hamilton, Ontario, Canada
Professor D. Tabak, George Mason University, Fairfax, Virginia, U.S.A.
Professor K. Valavanis, University of Southern Louisiana, Lafayette, U.S.A.
Multi-Arm Cooperating
Robots
Dynamics and Control
editedby
M.D. ZIVANOVIC
Robotics Center,
Mihajlo Pupin Institute, Belgrade,
Serbia and Montenegro
and
M.K. VUKOBRATOVIC
Robotics Center,
Mihajlo Pupin Institute, Belgrade,
Serbia and Montenegro
AC.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-10 1-4020-4268-X (HB)
ISBN-13 978-1-4020-4268-3 (HB)
ISBN-10 1-4020-4269-8 (e-book)
ISBN-13 978-1-4020-4269-0 (e-book)
Published by Springer,
P.O. Box 17, 3300 AADordrecht, The Netherlands.
www.springer.com
Printed on acid-free paper
All Rights Reserved
© 2006 Springer
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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or otherwise, without written permission from the Publisher, with the exception
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Printed in the Netherlands.
TABLE OF CONTENTS
LISTOFFIGURES ix
PREFACE xi
1. INTRODUCTIONTOCOOPERATIVEMANIPULATION 1
1.1 Cooperative Systems–Manipulation Systems 1
1.2 ContactintheCooperative Manipulation 4
1.3 TheNatureofContact 4
1.4 Introducing Coordinate Frames 7
1.5 GeneralConvention onSymbolsandQuantityDesignations 16
1.6 RelationtoContactTasksInvolving OneManipulator 18
2. PROBLEMSINCOOPERATIVEWORK 19
2.1 KinematicUncertainty 19
2.1.1Kinematicuncertainty duetomanipulator redundancy 19
2.1.2Kinematicuncertainty duetocontactcharacteristics 21
2.2 ForceUncertainty 22
2.3 SummaryofUncertainty ProblemsinCooperativeWork 24
2.4 TheProblemofControl 25
3. INTRODUCTIONTOMATHEMATICALMODELINGOF
COOPERATIVESYSTEMS 27
3.1 SomeKnownSolutions toCooperativeManipulation Models 28
3.2 AMethodtoModelCooperativeManipulation 30
3.3 Illustration oftheCorrectModelingProcedure 37
v
vi TableofContents
3.4 SimulationoftheMotionofaLinearCooperative System 51
3.5 SummaryoftheProblemofMathematical Modeling 54
4. MATHEMATICALMODELSOFCOOPERATIVESYSTEMS 57
4.1 Introductory Remarks 57
4.2 SettingUptheProblemofMathematicalModeling ofaComplex
Cooperative System 65
4.3 Theoretical BasesoftheModelingofanElasticSystem 66
4.4 ElasticSystemDeformations asaFunctionofAbsoluteCoordinates 74
4.5 ModelofElasticSystemDynamicsfortheImmobileUnloadedState 82
4.6 ModelofElasticSystemDynamicsforaMobileUnloadedState 86
4.7 Properties ofthePotentialEnergyandElasticityForceofthe
ElasticSystem 89
4.7.1 Propertiesofpotential energy andelasticity forceoftheelastic
systemintheloadedstatetranslation 91
4.7.2 Propertiesofpotential energy andelasticity forceoftheelastic
systemduringitsrotationintheloadedstate 94
4.8 ModelofManipulator Dynamics 100
4.9 KinematicRelations 101
4.10ModelofCooperativeSystemDynamicsfortheImmobile
UnloadedState 102
4.11ModelofCooperativeSystemDynamicsfortheMobile
UnloadedState 104
4.12FormsoftheMotionEquationsofCooperative System 106
4.13Stationary andEquilibrium StatesoftheCooperativeSystem 118
4.14Example 123
5. SYNTHESISOFNOMINALS 137
5.1 Introduction –ProblemDefinition 138
5.2 ElasticSystemNominals 142
5.2.1Nominalgripping oftheelasticsystem 142
5.2.2Nominalmotionoftheelasticsystem 153
5.3 NominalDrivingTorques 165
5.4 AlgorithmstoCalculatetheNominalMotioninCooperative
Manipulation 166
5.4.1 Algorithmtocalculatethenominalmotioningripping for
theconditions givenforthemanipulated objectMC 167
TableofContents vii
5.4.2 Algorithmtocalculatethenominalmotioningripping for
theconditions ofaselected contactpoint 168
5.4.3 Algorithmtocalculatethenominalgeneralmotionfor
theconditions givenforthemanipulated objectMC 171
5.4.4 Algorithmtocalculatethenominalgeneralmotionfor
theconditions givenforonecontactpoint 173
5.4.5 Exampleofthealgorithm fordetermining thenominalmotion 176
6. COOPERATIVESYSTEMCONTROL 189
6.1 Introduction totheProblemofCooperative SystemControl 189
6.2 Classification ofControlTasks 191
6.2.1 Basicassumptions 191
6.2.2 Classificationofthetasks 202
6.3 ChoiceofControlTasksinCooperativeManipulation 207
6.4 ControlLaws 212
6.4.1 Mathematicalmodel 212
6.4.2 Illustration oftheapplication oftheinputcalculation method 213
6.4.3 Controllawsfortracking thenominaltrajectory ofthe
manipulated objectMCandnominaltrajectories ofcontact
pointsofthefollowers 216
6.4.4 Behaviorofthenon-controlled quantities intrackingthe
manipulated objectMCandnominaltrajectories ofcontact
pointsofthefollowers 223
6.4.5 Controllawstotrackthenominaltrajectory ofthemanipulated
objectMCandnominalcontact forcesofthefollowers 229
6.4.6 Behaviorofthenon-controlled quantities intrackingthe
trajectory ofthemanipulated objectMCandnominalcontact
forcesofthefollowers 234
6.5 ExamplesofSelectedControlLaws 236
7. CONCLUSION:LOOKINGBACKONTHEPRESENTED
RESULTS 251
7.1 AnOverviewoftheIntroductory Considerations 251
7.2 OnMathematical Modeling 252
7.3 Cooperative SystemNominals 254
7.4 Cooperative SystemControlLaws 256
viii TableofContents
7.5 GeneralConclusions abouttheStudyofCooperativeManipulation 257
7.6 PossibleDirections ofFurtherResearch 258
APPENDIXA:ELASTICSYSTEMMODELFORTHEIMMOBILE
UNLOADEDSTATE 261
APPENDIXB:ELASTICSYSTEMMODELFORTHEMOBILE
UNLOADEDSTATE 269
REFERENCES 277
INDEX 283
LIST OF FIGURES
1 Cooperative manipulation system 3
2 Contact 6
3 Cooperative workofthefingersonanimmobileobject 8
4 Kinematicuncertainty duetocontact 22
5 Cooperative workoftwomanipulators ontheobject 23
6 Reducingthecooperative system toagrid 31
7 Approximation ofthecooperative systembyagrid 32
8 Linearelasticsystem 37
9 Approximating alinearelasticsystem 44
10 Blockdiagram ofthemodelofacooperative systemwithout
forceuncertainty 51
11 Resultsofsimulation ofa‘linear’elasticsystem 54
12 Elasticsystem 63
13 Displacements oftheelasticsystem nodes–thenotation system 66
14 Angulardisplacements oftheelasticsystem 76
15 Displacements oftheelasticsystem 78
16 Planardeformation oftheelasticsystem 83
17 Rotationoftheloadedelasticsystem 95
18 Blockdiagram ofthecooperative systemmodel 106
19 Elasticsystem oftwosprings 113
20 Initialposition ofthecooperative system 123
21a Simulationresultsforτj = 0,i,j = 1,2,3 127
i
21b Simulationresultsforτj = 0,i,j = 1,2,3 128
i
22a Simulationresultsforτ1 = 50[Nm]andτ1 = −50[Nm] 129
1 2
22b Simulationresultsforτ1 = 50[Nm]andτ1 = −50[Nm] 130
1 2
22c Simulationresultsforτ1 = 50[Nm]andτ1 = −50[Nm] 131
1 2
22d Simulationresultsforτ1 = 50[Nm]andτ1 = −50[Nm] 132
1 2
22e Simulationresultsforτ1 = 50[Nm]andτ1 = −50[Nm] 133
1 2
ix
x ListofFigures
22f Simulationresultsforτ1 = 50[Nm]andτ1 = −50[Nm] 134
1 2
22g Simulationresultsforτ1 = 50[Nm]andτ1 = −50[Nm] 135
1 2
23 Nominaltrajectory oftheobjectMC 143
24 Elasticdeviations fromthenominaltrajectory 146
25 Nominaltrajectory ofacontactpoint 163
26 ‘Linear’cooperative system 177
27 Nominalsforgripping amanipulated object 181
28 Nominalinputtoaclosed-loop cooperative system forgripping 182
29 Simulationresultsforgripping (open-loop cooperative system) 183
30 Nominalsformanipulated objectgeneralmotion 184
31 Nominalinputtoaclosed-loop cooperative system forgeneral motion 185
32 Simulationresultsformotion(open-loop cooperative system) 186
33 Mappingfromthedomainofinputstothedomainofstates 194
34 Mappingfromthedomainofstatestothedomainofinputs 195
35 Mappingfromthedomainofinputstothedomainofoutputs 195
36 Mappingfromthedomainofoutputs tothedomainofinputs 196
37 Mappingthrough thedomainofstates 196
38 Mappingofthecontrol systemdomain 197
39 Structureofthecontrol system 200
40 Mappingofthecontrol objectdomain 201
41 Mappingofthecooperative manipulation domain 205
42 Globalstructure oftheclosedloopsystem 215
43 Motionintheplaneoftheloadedelasticsystem 224
44 Blockdiagram oftheclosed-loop cooperative system 240
45a Gripping–tracking Y0 andY0 241
2 3
45b Gripping–tracking Y0 andY0 242
2 3
46a Gripping–tracking Y0 andF0 243
2 c2
46b Gripping–tracking Y0 andF0 244
2 c2
47a Generalmotion–tracking Y0 andY0 245
2 3
47b Generalmotion–tracking Y0 andY0 246
2 3
48a Generalmotion–tracking Y0 andF0 247
2 c2
48b Generalmotion–tracking Y0 andF0 248
2 c2